Pion form factors and decays. Introduction g-2 Transversity pion form factor:  models vs Lattice Conclusions A.E. Dorokhov in collaboration with W. Broniowski and E. Ruiz Arriola

Cosmology tell us that 95% of matter is not described in text-books yet Two search strategies: 1)High energy physics to excite heavy degrees of freedom. No any evidence till now. We live in LHC era! 2) Low energy physics to produce Rare processes in view of huge statistics. There are some rough edges of SM.  Muon anomaly  (g-2)  is the most famous example That’s intriguing

Gyromagnetic ratio Anomaly Some Definitions m μ = (94) MeV, m τ = (29) MeV m μ /m e = (54) PDG A charged particle with spin S has a magnetic moment 

The general form of the ff  vertex is F 1 is the electric charge distribution e l =eF 1 (0) F 2 corresponds to Anomalous Magnetic Moment (AMM) a l =(g l -2)/2=F 2 (0) F 3 corresponds to Anomalous Electric Dipole Moment d l =-e l /(2m l )F 3 (0) However, in SM a l is not zero due to Radiative Corrections d l =0 due to T- and P symmetries

Magnetic Anomaly QEDHadronicWeakSUSY or other new physics ? Basic of Standard Model

Electron anomaly is measured extremely accurately. QED test. It is the best for determining  For a lepton L, Mass Scale  contributes to a L as Tau anomaly is difficult to measure since its fast decay Muon anomaly is measured to 0.5 parts in a million (ppm) SM test. Thus muon AMM leads to a (m   m e ) 2 ~ enhancement of the sensitivity to New Physics versus the electron AMM. Lepton Anomalies

Electron AMM QED is at the level of the best theory ever built to describe nature To measurable level a e arises entirely from virtual electrons and photons The theoretical error is dominated by the uncertainty in the input value of the QED coupling α ≡ e 2 /(4π) Das ist fantastisch!

Tau anomaly Tau due to its highest mass is the best for searching for New Physics, But Tau is short living particle, so the precession method is not perspective The best existing limits <a  Exp <0.013 are obtained at OPAL, L3 and DELPHI (LEP, CERN) from the high energy process e + e -  e + e -  While the SM estimate is a  SM = (5) 10 -3

SM Contributions to Muon AMM from BNL From BNL E821 g-2 experiment ( ) From Standard Model A. Hoecker Tau2010 Update New Prop. E989 at Fermilab 0.14 ppm

plus the Hadronic Contribution estimated as The main question how to get such accuracy from theory. Kinoshita&Nio 2004, 2006 Czarnetski&Marciano&Vainshtein 2003 M. Davier, A. Hoecker, B. Malaescu, Z. Zhang 2010; F. Jegerlehner, Robert Szafron 2011

   h    he     LbL to g-2 Strong contributions to Muon AMMM Hadronic Vacuum polarization (Davier, Hoecker, Zhang) Hadronic Light-by-Light Scattering (Dubnicka, Bartos, Kuraev AED, Radzhabov, Zhevlakov)

(0) hadrons   W  e+e+ e –e – CVC: I =1 & V W: I =1 & V,A  : I =0,1 & V

Structure of hadronic LbL contribution Phenomenological and QCD Constraints are used to reduce Model Dependence

Interpretations SUSY m SUSY ≈ GeV Multi-Higgs Models Extra Dimensions<2TeV Dark Photons ∼ MeV, α’=10 -8 Light Higgs <10MeV? Davier etal 2010 g-2 is the most important constraint (for SUSY), even more important than dark matter

Since the pion has spin zero, its longitudinal spin structure in terms of quark and gluon degrees of freedom is trivial. An instructive quantity describing the transverse spin structure of hadrons is the probability density  (x,k tr,s tr ) In terms of moments one has the Generalized Form Factors (GFFs) Connection with Generalized Parton Distributions (GPDs)

The Generalized Form Factors of the Pion Introduce auxiliary vectors

Matrix Elements in the Chiral Models The Quark Propagator And the Quark-Pion vertex In the local (NJL) model one has

Transversity pion form factor (momentum space)

Transversity pion form factor (impact parameter space)

Forward: distribution function

Transverse size distribution function

Normalized density d T 2 (x)=b T 2 (x)/f(x)

 =log(1/x) One can interpret it as an evolution of the probability density for a stochastic motion of a particle in the transverse plane

QCD evolution GFFs B T evolve multiplicatively For Local Model one has for the Normalizations

The results: Lattice vs Local Model

The results: Lattice vs Nonlocal Models The transversity form factors in the chiral model (solid line) and in the instanton-motivated model (dashed line) for M q = 300 MeV.

Rare Pion Decay  0 →e + e -- from KTeV From KTeV E799-II EXPERIMENT at Fermilab experiment ( ) 99-00’ set, The result is based on observation of 794 candidate  0  e + e - events using K L  3  0 as a source of tagged  0 s. PRD (2007) One of the simplest process for THEORY

CLEO +QCD CLEO 3  diff What is next? It would be very desirable if Others will confirm KTeV result Also,    pair decay is very perspective AED, M. Ivanov PRD (2007)

The anomalous 511 keV  -ray signal from Galactic Center observed by INTEGRAL/SPI (2003) is naturally explained Enhancement in Rare Pion Decays from a Model of MeV Dark Matter (Boehm&Fayet ) was considered by Kahn, Schmitt and Tait (PRD 2008) excluded allowed

Conclusions High statistics Low-Energy experiments are important independent way to search for effects of New Physics, They complement to possible signals from High-Energy experiments (LHC,…) allowing to get combined restrictions on the parameters of hypothetical interactions They sensitive to New particles with low masses New experiments are urgent Theory has to be in good shape

Photon-pion transition form factor in nonperturbative QCD approach

The theory of hard exclusive processes was formulated within the factorization approach to perturbative quantum chromodynamics (pQCD) in The photon-pion transition γ*γ * →π⁰ is of special interest a)it is the simplest process for theory; b)related to the axial anomaly when both photons are real; c)At large photon virtualities it was studied in [6,7]

Photon-pion transition form factor: In the factorization approach Symmetric kinematics Asymmetric kinematics (measured by BABAR) Factorization f  J * 1/Q 2 and it settle in at ~ 1 GeV 2

BABAR disaster 2009 (and CELLO-1991, CLEO-1997) data Passive mode Active mode A. Data higher than BL Asymptotic limit, B. They GROW!

BABAR puzzles

Nonperturbative Nonlocal QCD Approach to  Form Factor And its asymptotic properties AED JETP Lett. (2003) Main properties The vertex F is equivalent of the light-cone pion WF

 representation F decays as 1/k 2 or faster The main property for asymptotic analysis  N.N.Bogolyubov, D.V. Shirkov O.I. Zavialov

Photon-pion transition form factor: Symmetric Kinematics Take chiral limit p 2 =0 and symmetric kinematics q 1 2 = q 2 2 = q 2  Leading Asymptotics as q 2  ∞ corresponds to γ  0 thus   ,  d  m  0, d   1+… Factorization – Yes!

Photon-pion transition form factor: Symmetric Kinematics  representation – URA! Factorization – URA!! Brodsky-Lepage - URA!!!

 Distribution Amplitude Pion DA is obtained if we perform substitution then AED JETP Lett. (2003)

 Distribution Amplitude For Instanton model For chiral model At x=0 one gets

 Distribution Amplitude

Take chiral limit p 2 =0 and symmetric kinematics q 1 2 = q 2, q 2 2 =0  Leading Asymptotics as q 2  ∞ corresponds to γ  0 or   0 thus   ,  F  m  0, F   1+…, but keep Exp small terms!    G m,0   0, G 0,m   g  m, G   F  Factorization – Not always! Photon-pion transition form factor: Asymmetric Kinematics Small  Small 

Factorization – Not at all! Small  Small  Small  Small  Photon-pion transition form factor: Asymmetric Kinematics

Photon-pion transition form factor: Asymmetric Kinematics in Instanton Model A) Confining Propagator B) Chiral Propagator

Photon-pion transition form factor: Asymmetric Kinematics in Chiral Model Thus one finds three possible Asymptotic regimes: 1)1/Q 2 Pion DA strongly suppressed at endpoints 2)“lnQ 2 /Q 2 ” Pion DA vanishes at endpoints, but almost flat otherwise 3)lnQ 2 /Q 2 Pion DA does not vanishes at endpoints Two last regimes may be responsible for BABAR data! BABAR puzzle is cracked!! Exact asymptotic expression It is nonfactorizable, no representation in terms of pion DA

M q = 300 MeV M q = 135 MeV

CONCLUSIONS 1. BABAR measured photon-pion form factor at large Q 2 in wide kinematical region and found that the data for Q 2 F(Q 2 ) exceed the asymptotic Brodsky-Lepage constant limit and, moreover, continue to growth – BABAR puzzle 2. We show that depending on the properties of the quark-pion vertex there are two possible shapes of the pion distribution amplitude: vanishing or not vanishing at the endpoints 3. These different cases provide different possibilities for the asymptotic behavior of the form factor Q 2 F(Q 2 ) ~ const Or Q 2 F(Q 2 ) ~ln(Q 2 ) 4. BABAR data, if will be confirmed, point out on specific properties of quark dynamics in the pion and of the underlying QCD vacuum

AntiBABAR Shock therapy 2009 A) B) C) Normalization by anomaly A “Factorization” B,C OPE B Value of Parameters M q,  and M and their meaning ? f  is external parameter, it is not defined All are based on flat (local) pion DA  (x)=1. How to justify that?? No QCD Evolution

AntiBABAR Shock therapy 2009 Nonfactorizable “Factorizable”

Chiral Model M q =125 MeV M q =135 MeVM q =115 MeV